Yes garlic can go bad eventually and make you sick if you eat the bad garlic.
Garlic is usually good for a few weeks to a month before it goes bad.
To keep the garlic from spoiling you should refrigerate or freeze the garlic and it will last longer.
Most times getting sick from the garlic comes from infusing the garlic with oil which then leads to botulism if the garlic is not infused with the oil properly.
Spoiled garlic forms brown spots on the cloves and turns from the usual white to a more yellow or brown color.
Good garlic should feel firm to the touch and if it's soft to the touch then it likely is bad.
Roasting garlic does not kill botulism.
Botulism is actually heat resistant so heat alone cannot kill the botulism.
Botulism with garlic is mostly a concern when infusing the garlic with oil and as long as you infuse the garlic with the olive oil properly then there should be no botulism.
To safely infuse garlic with olive oil you have to pressure can it.
Any other way of infusing the garlic with the olive oil has the potential for botulism to grow in the garlic and olive oil.
Garlic botulism issues relates to garlic in oil.
Oil creates the anaerobic environment botulism requires.
Garlic provides the water and the spores. Slow roasting kills the spores, if you hold at heat long enough and reduces the available water.
Keeping the oil in the fridge prevents growth too.
You can make the infused oil in small batches, in the oven on low, removing the cloves after a day or so and keeping the oil in the fridge, just to be extra safe.